The multivariate Hahn polynomials are constructed explicitly as the common eigenvectors of a family of second order difference operators. They are orthogonal with respect to the hypergeometric multinomial distribution. The main difference operator is adopted from the work of Karlin-McGregor in 1975. The minor ones are the subsets of the main one containing less and less variables. These operators commute with each other. In contrast to the multivariate Krawtchouk and Rahman like polynomials derived recently, the entire multivariate Hahn polynomials are rational functions of the system parameters. Complete sets of multivariate Krawtchouk and Meixner polynomials are derived by limiting procedures.